Chapter 6 Nonmonotonic Reasoning
Introduction
Classical logic is monotonic in the following sense: whenever a sentence A is a logical consequence of a set of sentences T, then A is also a consequence of an arbitrary superset of T. In other words, adding information never invalidates any conclusions.
Commonsense reasoning is different. We often draw plausible conclusions based on the assumption that the world in which we function and about which we reason is normal and as expected. This is far from being irrational. To the contrary, it is the best we can do in situations in which we have only incomplete information. However, as unexpected as it may be, it can happen that our normality assumptions turn out to be wrong. New information can show that the situation actually is abnormal in some respect. In this case we may have to revise our conclusions.
For example, let us assume that Professor Jones likes to have a good espresso after lunch in a campus cafe. You need to talk to her about a grant proposal. It is about 1:00 pm and, under normal circumstances, Professor Jones sticks to her daily routine. Thus, you draw a plausible conclusion that she is presently enjoying her favorite drink. You decide to go to the cafe and meet her there. As you get near the student center, where the cafe is located, you see people streaming out of the building. One of them tells you about the fire alarm that just went off. The new piece of information invalidates the normality assumption and so the conclusion about the present location of Professor Jones, too.
Such reasoning, where additional information may invalidate conclusions, is called nonmonotonic. It has been a focus of extensive studies by the knowledge representation community since the early eighties of the last century. This interest was fueled by several fundamental challenges facing knowledge representation such as modeling and reasoning about rules with exceptions or defaults, and solving the frame problem.
Most rules we use in commonsense reasoning—like university professors teach, birds fly, kids like ice-cream, Japanese cars are reliable—have exceptions. The rules describe what is normally the case, but they do not necessarily hold without exception. This is obviously in contrast with universally quantified formulas in first order logic. The sentence simply excludes the possibility of non-teaching university professors and thus cannot be used to represent rules with exceptions. Of course, we can refine the sentence to However, to apply this rule, say to Professor Jones, we need to know whether Professor Jones is exceptional (for instance, professors who are department Chairs do not teach). Even if we assume that the unary predicate can be defined precisely, which is rarely the case in practice as the list of possible exceptions is hard—if not impossible—to complete, we will most often lack information to derive that Professor Jones is not exceptional. We want to apply the rule even if all we know about Dr. Jones is that she is a professor at a university. If we later learn she is a department Chair—well, then we have to retract our former conclusion about her teaching classes. Such scenarios can only be handled with a nonmonotonic reasoning formalism.
To express effects of actions and reason about changes in the world they incur, one has to indicate under what circumstances a proposition whose truth value may vary, a fluent, holds. One of the most elegant formalisms to represent change in logic, situation calculus[88], [89], [112], uses situations corresponding to sequences of actions to achieve this. For instance, the fact that Fred is in the kitchen after walking there, starting in initial situation , is represented as The predicate holds allows us to state that a fluent, here , holds in a particular situation. The expression is an action, and the expression is the situation after Fred walked to the kitchen, while in situation .
In situation calculus, effects of actions can easily be described. It is more problematic, however, to describe what does not change when an event occurs. For instance, the color of the kitchen, the position of chairs, and many other things remain unaffected by Fred walking to the kitchen. The frame problem asks how to represent the large amount of non-changes when reasoning about action.
One possibility is to use a persistence rule such as: what holds in a situation typically holds in the situation after an action was performed, unless it contradicts the description of the effects of the action. This rule is obviously nonmonotonic. Just adding such a persistence rule to an action theory is not nearly enough to solve problems arising in reasoning about action (see Chapters 16–19Chapter 16Chapter 17Chapter 18Chapter 19 in this volume). However, it is an important component of a solution, and so the frame problem has provided a major impetus to research of nonmonotonic reasoning.
Handling rules with exceptions and representing the frame problem are by no means the only applications that have been driving research in nonmonotonic reasoning. Belief revision, abstract nonmonotonic inference relations, reasoning with conditionals, semantics of logic programs with negation, and applications of nonmonotonic formalisms as database query languages and specification languages for search problems all provided motivation and new directions for research in nonmonotonic reasoning.
One of the first papers explicitly dealing with the issue of nonmonotonic reasoning was a paper by Erik Sandewall [115] written in 1972 at a time when it was sometimes argued that logic is irrelevant for AI since it is not capable of representing nonmonotonicity in the consequence relation. Sandewall argued that it is indeed possible, with a moderate modification of conventional (first order) logic, to accommodate this requirement. The basic idea in the 1972 paper is to allow rules of the form where, informally, C can be inferred if A was inferred and B cannot be inferred. The 1972 paper discusses consequences of the proposed approach, and in particular it identifies that it leads to the possibility of multiple extensions. At about the same time Hewitt published his work on Planner [55], where he proposed using the thnot operator for referring to failed inference.
In this chapter we give a short introduction to the field. Given its present scope, we do not aim at a comprehensive survey. Instead, we will describe three of the major formalisms in more detail: default logic in Section 6.2, autoepistemic logic in Section 6.3, and circumscription in Section 6.4. We will then discuss connections between these formalisms. It is encouraging and esthetically satisfying that despite different origins and motivations, one can find common themes.
We chose default logic, autoepistemic logic, and circumscription for the more detailed presentation since they are prominent and typical representatives of two orthogonal approaches: fixed point logics and model preference logics. The former are based on a fixed point operator that is used to generate—possibly multiple—sets of acceptable beliefs (called extensions or expansions), taking into account certain consistency conditions. Nonmonotonicity in these approaches is achieved since what is consistent changes when new information is added. Model preference logics, on the other hand, are concerned with nonmonotonic inference relations rather than formation of belief sets. They select some preferred or normal models out of the set of all models and define nonmonotonic inference with respect to these preferred (normal) models only. Here nonmonotonicity arises since adding new information changes the set of preferred models: models that were not preferred before may become preferred once we learn new facts.
Preference logics and their generalizations are important not only as a broad framework for circumscription. They are also fundamental for studies of abstract nonmonotonic inference relations. In Section 6.5, we discuss this line of research in more detail and cover such related topics as reasoning about conditionals, rational closure, and system Z.
In the last section of the chapter, we discuss the relationship between the major approaches, and present an overview of some other research directions in nonmonotonic reasoning. By necessity we will be brief. For a more extensive treatment of nonmonotonic reasoning we refer the reader to the books (in order of appearance) [2], [11], [16], [17], [25], [43], [78], [80], [85].
Section snippets
Default Logic
Default reasoning is common. It appears when we apply the Closed-World Assumption to derive negative information, when we use inference rules that admit exceptions (rules that hold under the normality assumption), and when we use frame axioms to reason about effects of actions. Ray Reiter, who provided one of the most robust formalizations of default reasoning, argued that understanding default reasoning is of foremost importance for knowledge representation and reasoning. According to Reiter
Autoepistemic Logic
In this section, we discuss autoepistemic logic, one of the most studied and influential nonmonotonic logics. It was proposed by Moore in [92], [93] in a reaction to an earlier modal nonmonotonic logic of McDermott and Doyle [91]. Historically, autoepistemic logic played a major role in the development of nonmonotonic logics of belief. Moreover, intuitions underlying autoepistemic logic and studied in [47] motivated the concept of a stable model of a logic program [49]3
Motivation
Circumscription was introduced by John McCarthy [86], [87]. Many of its formal aspects were worked out by Vladimir Lifschitz who also wrote an excellent overview [74]. We follow here the notation and terminology used in this overview article.
The idea underlying circumscription can be explained using the teaching professors example discussed in the introduction. There we considered using the following first order formula to express professors normally teach:
Nonmonotonic Inference Relations
Having discussed three specific nonmonotonic formalisms in considerable detail, we will now move on to an orthogonal theme in nonmonotonic reasoning research: an abstract study of inference relations associated with nonmonotonic (defeasible) reasoning. Circumscription fits in this theme quite well—it can be viewed as an example of a preferential model approach, yielding a preferential inference relation. However, as we mention again at the end of this chapter, it is not so for default and
Further Issues and Conclusion
In this section we discuss the relationship between the major approaches we presented earlier. We first relate default logic and autoepistemic logic (Section 6.6.1), then default logic and circumscription (Section 6.6.2). Finally, we give pointers to some other approaches which we could not present in more detail in this chapter (Section 6.6.3).
Acknowledgements
We would like to thank Vladimir Lifschitz and Hudson Turner for helpful comments and suggestions.
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